Constructions of vector-valued filters and vector-valued wavelets (Q1760555)

Summary: Let \({\mathbf a}= (a_1, a_2, \dots, a_m) \in \mathbb C^m\) be an \(m\)-dimensional vector. Then, it can be identified with an \(m \times m\) circulant matrix. By using the theory of matrix-valued wavelet analysis [\textit{A. T. Walden} and \textit{A. Serroukh} [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 458, No. 2017, 157--179 (2002; Zbl 1010.62087)], we discuss the vector-valued multiresolution analysis. Also, we derive several different designs of finite length of vector-valued filters. The corresponding scaling functions and wavelet functions are given. Specially, we deal with the construction of filters on symmetric matrix-valued functions space.